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 Incoming links: Electric field line

Use

x = ρ cos ϕ

y = ρ sin ϕ

V = A (x^{2} - y^{2})

E = (- 2 A x, 2 A y, 0),

E_{ρ} = - 2 A ρ cos 2 ϕ, E_{ϕ} = 2 A ρ sin 2 ϕ, E_{z} = 0.

For

A > 0

, arrows point toward the origin along the

x

axis and away from it along the

y

axis. The off-axis electric field lines satisfy

\frac{d y}{d x} = \frac{E _{y}}{E _{x}} = - \frac{y}{x}, x y = constant .

The axes themselves are the separating field lines; the origin is a field null. Reversing

A

reverses every arrow. The field is independent of

z

, and

A

has units

V m^{- 2}